Choosing a vacation location based on climate at a given time of the year

For a given time of year, and a given ideal set of weather conditions, how do you find out where in the world is likely to have them?

Here is where some knowledge of geography and general weather conditions comes in handy, as it gives you a way to narrow down the search. For example, anywhere close to the equator will be fairly constant temperature year-round, while places north of it will be in spring roughly April-June, in summer roughly July-August, in fall roughly September-November, and in winter roughly December-March. The further north, the shorter and less extreme the summer, but the longer and more extreme the winter. Everything is reversed if you go south of the equator.

You can then look at a globe, figure out the time of year you want to travel, the kind of weather you desire, and determine bands north or south of the equator likely to have that weather at that time of the year.

Obviously, this is a very crude and rough tool, but it gives you a way to focus your search on a smaller area. You can then look at area-specific factors. For example, for the same latitude in Southern California, the temperature is likely to change significantly as you move even small increments east or west. It might be 72F during the day near the coast, but 90F during the day and 60F at night just a few miles inland.

I would like to take some time off in August (school holidays), and I would like to find counties or regions that usually have reasonably nice weather

Depending on how far you wish you travel, you might consider places such as Stockholm, Sweden (in Europe but not as crowded as Rome or Paris), Cape Town or Johannesburg in South Africa (their winter, but those are fairly temperate so the weather should be pleasant), Curaçao (where it is 80F year-around), etc.